Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C32×#117 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C22 |
| commutator | C2 | C22×C32×C4 |
| Frattini, cosocle | C22 | C23×C32 |
| socle | C22×C32 | C23 |
| prime radical | C23×C32 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ G | |
228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,15);(6,9)(10,12)(16,17,18);(5,7,8,11)(6,10,9,12)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 814);